# Context-free language

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In formal language theory, a context-free language (CFL) is a language generated by some context-free grammar (CFG). Different CF grammars can generate the same CF language, or conversely, a given CF language can be generated by different CF grammars. It is important to distinguish properties of the language (intrinsic properties) from properties of a particular grammar (extrinsic properties).

The set of all context-free languages is identical to the set of languages accepted by pushdown automata, which makes these languages amenable to parsing. Indeed, given a CFG, there is a direct way to produce a pushdown automaton for the grammar (and corresponding language), though going the other way (producing a grammar given an automaton) is not as direct.

Context-free languages have many applications in programming languages; for example, the language of all properly matched parentheses is generated by the grammar $S\to SS ~|~ (S) ~|~ \varepsilon$. Also, most arithmetic expressions are generated by context-free grammars.

## Examples

An archetypical context-free language is $L = \{a^nb^n:n\geq1\}$, the language of all non-empty even-length strings, the entire first halves of which are $a$'s, and the entire second halves of which are $b$'s. $L$ is generated by the grammar $S\to aSb ~|~ ab$, and is accepted by the pushdown automaton $M=(\{q_0,q_1,q_f\}, \{a,b\}, \{a,z\}, \delta, q_0, z, \{q_f\})$ where $\delta$ is defined as follows:

$\delta(q_0, a, z) = (q_0, az)$
$\delta(q_0, a, a) = (q_0, aa)$
$\delta(q_0, b, a) = (q_1, \varepsilon)$
$\delta(q_1, b, a) = (q_1, \varepsilon)$

$\delta(\mathrm{state}_1, \mathrm{read}, \mathrm{pop}) = (\mathrm{state}_2, \mathrm{push})$

Unambiguous CFLs are a proper subset of all CFLs: there are inherently ambiguous CFLs. An example of an inherently ambiguous CFL is the union of $\{a^n b^m c^m d^n | n, m > 0\}$ with $\{a^n b^n c^m d^m | n, m > 0\}$. This set is context-free, since the union of two context-free languages is always context-free. But there is no way to unambiguously parse strings in the (non-context-free) subset $\{a^n b^n c^n d^n | n > 0\}$ which is the intersection of these two languages.[1]

## Languages that are not context-free

The set $\{a^n b^n c^n d^n | n > 0\}$ is a context-sensitive language, but there does not exist a context-free grammar generating this language.[1] So there exist context-sensitive languages which are not context-free. To prove that a given language is not context-free, one may employ the pumping lemma for context-free languages or a number of other methods, such as Ogden's lemma or Parikh's theorem.[2]

## Closure properties

Context-free languages are closed under the following operations. That is, if L and P are context-free languages, the following languages are context-free as well:

• the union $L \cup P$ of L and P
• the reversal of L
• the concatenation $L \cdot P$ of L and P
• the Kleene star $L^*$ of L
• the image $\varphi(L)$ of L under a homomorphism $\varphi$
• the image $\varphi^{-1}(L)$ of L under an inverse homomorphism $\varphi^{-1}$
• the cyclic shift of L (the language $\{vu : uv \in L \}$)

Context-free languages are not closed under complement, intersection, or difference. However, if L is a context-free language and D is a regular language then both their intersection $L\cap D$ and their difference $L\setminus D$ are context-free languages.

### Nonclosure under intersection and complement

The context-free languages are not closed under intersection. This can be seen by taking the languages $A = \{a^n b^n c^m \mid m, n \geq 0 \}$ and $B = \{a^m b^n c^n \mid m,n \geq 0\}$, which are both context-free.[3] Their intersection is $A \cap B = \{ a^n b^n c^n \mid n \geq 0\}$, which can be shown to be non-context-free by the pumping lemma for context-free languages.

Context-free languages are also not closed under complementation, as for any languages A and B: $A \cap B = \overline{\overline{A} \cup \overline{B}}$.

## Decidability properties

The following problems are undecidable for arbitrary context-free grammars A and B:

• Equivalence: Given two context-free grammars A and B, is $L(A)=L(B)$?
• Intersection Emptiness: Given two context-free grammars A and B, is $L(A) \cap L(B) = \emptyset$ ? However, the intersection of a context-free language and a regular language is context-free,[4] and the variant of the problem where B is a regular grammar is decidable.
• Containment: Given a context-free grammar A, is $L(A) \subseteq L(B)$ ? Again, the variant of the problem where B is a regular grammar is decidable.
• Universality: Given a context-free grammar A, is $L(A)=\Sigma^*$ ?

The following problems are decidable for arbitrary context-free languages:

• Emptiness: Given a context-free grammar A, is $L(A)=\emptyset$ ?
• Finiteness: Given a context-free grammar A, is $L(A)$ finite?
• Membership: Given a context-free grammar G, and a word $w$, does $w \in L(G)$ ? Efficient polynomial-time algorithms for the membership problem are the CYK algorithm and Earley's Algorithm.

## Parsing

Determining an instance of the membership problem; i.e. given a string $w$, determine whether $w \in L(G)$ where $L$ is the language generated by some grammar $G$; is also known as parsing.

Formally, the set of all context-free languages is identical to the set of languages accepted by pushdown automata (PDA). Parser algorithms for context-free languages include the CYK algorithm and the Earley's Algorithm.

A special subclass of context-free languages are the deterministic context-free languages which are defined as the set of languages accepted by a deterministic pushdown automaton and can be parsed by a LR(k) parser.[5]

See also parsing expression grammar as an alternative approach to grammar and parser.

## Notes

1. ^ a b
2. ^ How to prove that a language is not context-free?
3. ^ A context-free grammar for the language A is given by the following production rules, taking S as the start symbol: SSc | aTb | ε; TaTb | ε. The grammar for B is analogous.
4. ^ Salomaa (1973), p. 59, Theorem 6.7
5. ^ Knuth, Donald E. (July 1965). "On the translation of languages from left to right". Information and Control 8 (6): 607–639. doi:10.1016/S0019-9958(65)90426-2. Retrieved 29 May 2011. edit

## References

• Seymour Ginsburg (1966). The Mathematical Theory of Context-Free Languages. New York, NY, USA: McGraw-Hill, Inc.
• Hopcroft, John E.; Ullman, Jeffrey D. (1979). Introduction to Automata Theory, Languages, and Computation (1st ed.). Addison-Wesley.
• Arto Salomaa (1973). Formal Languages. ACM Monograph Series.
• Michael Sipser (1997). Introduction to the Theory of Computation. PWS Publishing. ISBN 0-534-94728-X. Chapter 2: Context-Free Languages, pp. 91–122.
• Jean-Michel Autebert, Jean Berstel, Luc Boasson, Context-Free Languages and Push-Down Automata, in: G. Rozenberg, A. Salomaa (eds.), Handbook of Formal Languages, Vol. 1, Springer-Verlag, 1997, 111-174.